Matching polytons

Hladky, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs. Combinatorial optimization studies the structure of the matching polytope and the fractional vertex cover polytope of a graph. Here, in analogy, we initiate the study of the structure of the set of all matchings and of all fractional vertex covers in a graphon. We call these sets the matching polyton and the fractional vertex cover polyton. We also study properties of matching polytons and fractional vertex cover polytons along convergent sequences of graphons. As an auxiliary tool of independent interest, we prove that a graphon is $r$-partite if and only if it contains no graph of chromatic number $r+1$. This in turn gives a characterization of bipartite graphons as those having a symmetric spectrum.


Introduction
Theories of graph limits are arguably one of the most important directions in discrete mathematics in the last decade. They link graph theory to analytic parts of mathematics, and through this connection introduce new tools to graph theory. With regard to such applications, the most fruitful theory has been that of flag algebras, [13]. Here, we deal with the theory of graphons developed by Borgs, Chayes, Lovász, Szegedy, Sós, and Vesztergombi, [1,11]. This theory, too, has found numerous applications in extremal graph theory (e.g., [9]), theory of random graphs [2], and in our understanding of properties of Szemerédi regularity partitions, see e.g. [12]. Much of the theory is built on counterparts of concepts well-known from the world of finite graphs (such as subgraph densities or cuts). Ideally, such counterparts are continuous with respect to the cut-distance, and are equal to the original concept when the graphon in question is a representation of a finite graph.
Hladký, Hu, and Piguet in [7] translated the concept of vertex-disjoint copies of a fixed finite graph F in a (large) host graph to graphons. Following preceding literature on this topic, they use the name F -tiling (in a graph or in a graphon). This allows them to introduce the F -tiling ratio of a graphon. They also translate the closely related concept of (fractional) F -covers in finite graphs to graphons which is a dual concept to F -tilings. The case when An extended abstract describing these results will appear in the proceedings of the EuroComb2017 conference, [4].
F is an edge, F = K 2 , is the most important. Then F -tilings are exactly matchings, the F -tiling ratio is just the matching ratio 1 , and (fractional) F -covers are exactly (fractional) vertex covers.
In this paper we deal exclusively with the case F = K 2 and from now on we specialize our description to this case. Some of our results, however, generalize to other F -tilings. We discuss the possible generalizations in Section 5.4.
Hladký, Hu, and Piguet give transference statements between the finite (i.e., graph) and limit (i.e., graphon) versions of the said notions. They mostly study the numerical quantities provided by the theory they develop, that is, the matching ratio and the fractional vertex cover ratio. Given a graphon W , we denote these two quantities (which we define in Section 2.3) by match(W ) and fcov(W ). One of the main results from [7] is the counterpart of the prominent linear programming duality between the fractional matching number of a graph and its fractional vertex cover number. Since -as Hladký, Hu, and Piguet argue -in the graphon world there is no distinction between matchings and fractional matchings, their LP duality has the form match(W ) = fcov(W ). In [7] and [6] they give applications of this LP duality in extremal graph theory. 2 In this paper, on the other hand, we study the sets of all fractional matchings, and of all fractional vertex covers. In the case of a finite graph G, these sets are known as the fractional matching polytope and the fractional vertex cover polytope. We shall denote them by MATCH(G) and FCOV(G). Study of MATCH(G) and FCOV(G) (and study of related polytopes such as the (integral) matching polytope and the perfect matching polytope) is central in polyhedral combinatorics and in combinatorial optimization. From the numerous results on the geometry of these polytopes, let us mention integrality of the fractional matching polytope and fractional vertex cover polytope of a bipartite graph, or the Edmonds' perfect matching polytope theorem. Here, we initiate a parallel study in the context of graphons. While in the finite case we have MATCH(G) ⊆ R E(G) and FCOV(G) ⊆ R V (G) , given a graphon W : Ω 2 → [0, 1], for the corresponding objects MATCH(W ) and FCOV(W ) it turns out that we have MATCH(W ) ⊆ L 1 (Ω 2 ) and FCOV(W ) ⊆ L ∞ (Ω). So, while MATCH(G) and FCOV(G) is studied using tools from linear algebra, in order to study MATCH(W ) and FCOV(W ) we need to use the language of functional analysis. We employ the -on word ending used among others for graphons and for permutons and call the limit counterparts to polytopes (such as MATCH(W ) and FCOV(W )) polytons.
1.1. Overview of the paper. In Section 2 we recall the necessary background concerning graphons and the theory of matchings/tilings in graphons developed in [7].
In Section 3 we treat (half)-integrality of the extreme points of the fractional vertex cover polyton of a graphon. As an application, we deduce a graphon version of the Erdős-Gallai theorem on matchings in dense graphs. In Section 4 we show that if a sequence of graphons (W n ) n converges to a graphon W then "MATCH(W n ) asymptotically contains MATCH(W )". This result is dual to results from [7] on the relation between FCOV(W n ) and FCOV(W ), which we recall in Section 4.1.
Section 5 contains some concluding remarks.
Lemma 1. Let (Ω, ν) be a probability space and let A, B ⊆ Ω be given. Let D ⊆ A × B be a set of positive ν ⊗2 measure. Then for every ε > 0 there is a measurable rectangle Proof. Let us fix ε > 0. By the definition of the product measure, we can find measurable Then there is a natural number m such that Now the finite union m i=1 R i can obviously be decomposed into finitely many pairwise disjoint measurable rectangles S 1 , . . . , S l . Then the inequality (1) can be rewritten as Thus there is some i ∈ {1, . . . , l} such that ν ⊗2 (S i \ D) < εν ⊗2 (S i ). The corresponding S i is the wanted measurable rectangle R.
2.2. Graphon basics. Our notation follows [10]. Throughout the paper we shall assume that Ω is an atomless Borel probability space equipped with a measure ν (defined on an implicit σ-algebra). We denote by ν ⊗k the product measure on Ω k . Let us recall that a graphon W : Suppose that F is a graph on vertex set [k]. Then the density of F in a graphon W is defined as t(F, W ) = Recall that the cut-norm · and the cut-distance dist (·, ·) are defined by where the infimum in the definition of the cut-distance ranges over all measure-preserving bijections on Ω, and W φ is defined by W φ (x, y) = W (φ(x), φ(y)).

Introducing matchings and vertex covers in graphons.
We introduce the notion of matchings in a graphon. Our definitions follow [7], where they were given in the more general context of F -tilings.
supp m ⊆ supp W up to a null-set, and (3) for almost every x ∈ Ω, we have y m(x, y) + y m(y, x) ≤ 1.
In [7] we argued in detail why this is "the right" notion of matchings. We do not repeat this discussion here and only briefly mention that the requirements in Definition 2 are counterparts to fractional matchings in finite graphs. Namely, a fractional matching in a graph G can be represented as a function f : V (G) 2 → R such that (1) f ≥ 0, (2) if f (x, y) > 0 then xy ∈ E(G), and (3) for every x ∈ V (G), we have y f (x, y) + y f (y, x) ≤ 1.
(Note that usually fractional matchings are represented using symmetric functions. This is however only a notational matter. 3 ) Remark 3. As said already in the Introduction, even though Definition 2 is inspired by fractional matchings in finite graphs, the resulting graphon concept is referred to as "matchings". This is because in the graphon world every function m from Definition 2 behaves in many ways as an integral matching.
Given a matching m in a graphon W we define its size, m = x y m(x, y). The matching ratio of W , denoted by match(W ), is defined as the supremum of the sizes of all matchings in W .
We write MATCH(W ) ⊆ L 1 (Ω 2 ) for the set of all matchings in W . It is straightforward to check that this set is convex (like the set of fractional matchings in a finite graph) and closed (if we consider the norm topology on L 1 (Ω 2 )). But -unlike the finite case -it need not be compact. To see this consider the graphon U : [0, 1] 2 → [0, 1] defined as U (x, y) = 1 for x + y ≤ 1 and U (x, y) = 0 for x + y > 1. This example was first given in [7] in a somewhat different context. For ε positive, consider a matching m ε defined to be 1/(2ε) on a stripe of width ε along the diagonal x + y = 1 and zero otherwise. This is shown on Figure 1. It is clear that the matchings m ε do not contain any convergent subsequence, as we let ε → 0 + . Considering the weak topology on the space L 1 (Ω 2 ) (that is the topology generated by the dual space L ∞ (Ω 2 )) does not help as the same counterexample easily shows. Therefore considering the set MATCH(W ) as a subset of the second dual of L 1 (Ω 2 ) equipped with its weak * topology seems to be the only reasonable way to have a natural compactification of MATCH(W ). However, we did not need go that far.
We can now proceed with the definition of fractional vertex covers of a graphon. First, recall that a function c : V (G) → [0, 1] is a fractional vertex cover of a finite graph G if we have c(x) + c(y) ≥ 1 for each xy ∈ E(G). Thus, the graphon counterpart is as follows.
Definition 4. Suppose that W : Ω 2 → [0, 1] is a graphon. We say that a function c ∈ L ∞ (Ω) is a fractional vertex cover of W if 0 ≤ c ≤ 1 almost everywhere and the set has measure 0.
A fractional vertex cover is called an integral vertex cover if its values are from the set {0, 1} almost everywhere.
Given a fractional vertex cover c of a graphon W we define its size, c = x c(x). The fractional cover number, fcov(W ) is the infimum of sizes of all fractional vertex covers of W .
We write FCOV(W ) ⊆ L ∞ (Ω) for the set of all fractional vertex covers of W . It is straightforward to check that this set is convex. Further, as was first shown in [7, Theorem 3.14], it is also compact in the space L ∞ (Ω) equipped with the weak * topology.

Graphon counterparts.
Suppose that L is a vector space, and suppose that X ⊆ L is a convex set. Recall that a point x ∈ X is called an extreme point of X if the only pair We shall write E(X) to denote the set of all extreme points of X.
When L is finite-dimensional and X is a polytope in L then the extreme points of X are exactly its vertices. The importance of the notion of extreme points comes from the Krein-Milman theorem which states that in a locally convex topological vector space, each compact convex set equals to the closed convex hull of its extreme points.
Thus, the graphon counterparts to the results described in Section 3.1 will be expressed in terms of E(FCOV(W )). Let us now state these counterparts. The notion of extreme points is not the only generalization of vertices of a polytope. Another basic notion from convex analysis is that of exposed points. Its stronger variant, the notion of strongly exposed points, can be used for a characterization of the Radon-Nikodym property of Banach spaces which is an extensively studied topic. A point x in a convex set X is exposed if there exists a continuous linear functional for which x attains its strict maximum on X.
It is easy to see that every exposed point is extreme. The converse does not hold; a wellknown counterexample in R 2 is shown in Figure 2. We see, however, that every extreme point of the fractional vertex cover polyton of a bipartite graphon is exposed. Indeed, let φ ∈ E(FCOV(W )) for some bipartite graphon W : Ω 2 → [0, 1]. Then Theorem 5 tells us that the sets A = φ −1 (0) and B = φ −1 (1) partition Ω. It is now clear that the linear functional f , is strictly maximized at φ on FCOV(W ). We leave it as an open question whether every extreme point of the fractional vertex cover polyton is also exposed even for non-bipartite graphons.
3.3. Proof of Theorem 5. For the proof of Theorem 5 we shall need the following easy fact.
Proof of Theorem 5. Let Ω = Ω A ⊔ Ω B be a partition into two sets of positive measure such that W is zero almost everywhere on ( is not integral. Using the notation from Fact 8, define two functions c ′ , c ′′ : As c is not integral, we have that c is distinct from c ′ and c ′′ . We conclude that c is not an extreme point of FCOV(W ).

3.4.
Proof of Theorem 6. The proof of Theorem 6 is very similar to that of Theorem 5. We first state the counterpart of Fact 8 we need to this end. We omit the proof as it is almost the same as that of Fact 8.
Proof of Theorem 6. Suppose that c ∈ FCOV(W ) is not half-integral. Consider the sets Ω A = {x ∈ Ω : 0 ≤ c(x) ≤ 1 2 } and Ω B = {x ∈ Ω : 1 2 < c(x) ≤ 1}. Using the notation from Fact 9, define two functions c ′ , c ′′ : for each a ∈ Ω A and b ∈ Ω B . By Fact 9, we have that As c is not half-integral, we have that c is distinct from c ′ and c ′′ . We conclude that c is not an extreme point of FCOV(W ).
3.5. Proof of Theorem 7. Lemmas 10 and 11 are key for proving Theorem 7. These lemmas (and the generalization of Lemma 10 given in Proposition 21) may be of independent interest.
Lemma 10. Suppose that W : Ω 2 → [0, 1] is a graphon. Then W is bipartite if and only if for every odd integer k ≥ 3 it holds t(C k , W ) = 0.
Proof. Suppose first that there is an odd integer k ≥ 3 such that t(C k , W ) > 0. Let Ω = Ω 0 ⊔ Ω 1 be an arbitrary decomposition of Ω into two disjoint measurable subsets. Then there exists (i j ) k j=1 ∈ {0, 1} k such that As k is odd, there is j ∈ {1, . . . , k} such that i j = i j+1 (here we use the cyclic indexing, i.e. k + 1 = 1). By Fubini's theorem in other words Ω 2 i W > 0 for some i ∈ {0, 1}. As the decomposition Ω = Ω 0 ⊔ Ω 1 was chosen arbitrarily, this proves that W is not bipartite. Now suppose that t(C k , W ) = 0 for every odd integer k ≥ 3. By transfinite induction we define a transfinite sequence {(A α , B α ) : α ≤ γ} (for some countable ordinal γ) consisting of pairs of measurable subsets of Ω such that Once we are done with the construction, the bipartiteness of W immediately follows by the equation ν(A γ ∪ B γ ) = 1 together with (iii) and (v).
By Fubini's theorem, we easily conclude that z ∈ C ℓ+1 . But this contradicts the fact that z / ∈ A ∪ B.
To finish the proof, it suffices to observe that for some countable ordinal γ we get ν(A γ ∪ B γ ) = 1, and then the construction stops.
Lemma 10 is a graphon counterpart to the well-known fact that a graph is bipartite if and only if it does not contain odd-cycles.
Lemma 11. Suppose that W : Ω 2 → [0, 1] is a graphon. If W is not bipartite then there exists an odd integer k ≥ 3 with the following property. For each ε > 0 there exist pairwise disjoint sets A 1 , . . . , A k ⊆ Ω of the same positive measure α, such that for each h ∈ [k], W is positive everywhere on A h × A h+1 except a set of measure at most εα 2 . Here, we use cyclic indexing, A k+1 = A 1 .
Proof. Suppose that W is not bipartite. By Lemma 10 there is an odd integer k ≥ 3 such that We find a natural number n such that We fix a decomposition Ω = n i=1 Ω i of Ω into pairwise disjoint sets of the same measure 1 n . We also set D = (x 1 , . . . , x k ) ∈ Ω k : there are i, j ∈ {1, . . . , k} and ℓ ∈ {1, . . . , n} Then we have and so By (3) and (6), we get By this and (5) there are pairwise distinct integers ℓ 1 , . . . , ℓ k ∈ {1, . . . , n} such that and so the set is of positive measure. Now let us fix ε > 0, and let δ > 0 be such that Recall that the σ-algebra of all measurable subsets of Ω ℓ 1 × . . . × Ω ℓ k is generated by the algebra consisting of all finite unions of measurable rectangles. Thus there is a finite union Without loss of generality, we may assume that the measurable rectangles R 1 , . . . , R m are pairwise disjoint. Then we have Now the left-hand side of (10) can be expressed as i.e. as a convex combination of ν ⊗k (E∩R i ) ν ⊗k (R i ) , i = 1, . . . , m. Therefore by (10), there is an index Let R i 0 be of the form For every i = 1, . . . , k, we fix a finite decomposition disjoint sets, such that ν(B 0 i ) ≤ 1 p and ν(B j i ) = 1 p for j = 1, . . . , q i . Then we clearly have and so The left-hand side of (14) can be expressed as the following convex combination: Therefore by (14), there are j i ∈ {1, . . . , q i }, i = 1, . . . , k, such that We set A i = B j i i for i = 1, . . . , k. Then A 1 , . . . , A k are pairwise disjoint (as A i ⊆ B i ⊆ Ω ℓ i for every i), and each of these sets has the same measure α = 1 p . By (8) we have (16) By (15), Plugging this into (16), we get as required.
Proof of Theorem 7. We shall prove the counterpositive. Suppose that W is not bipartite. Let COV(W ) be the closure (in the weak * topology) of the convex hull of all integral vertex covers of W . Clearly, we have COV(W ) ⊆ FCOV(W ), and each integral vertex cover of FCOV(W ) is contained in COV(W ). Below, we shall show that The Krein-Milman Theorem then tells us that E FCOV(W ) \ COV(W ) = ∅. It will thus follow that there exists a non-integral fractional vertex cover in E FCOV(W ) , as was needed to show.
Take c : Ω → [0, 1] to be constant 1 2 . Clearly, c ∈ FCOV(W ). In order to show (17), it suffices to prove that c / ∈ COV(W ). Let k be the odd integer given by Lemma 11. Let ε = 1 32k 2 , and let the sets A 1 , . . . , A k of measure α > 0 be given by Lemma 11. In order to prove that c is not in the weak * closure of the convex hull of integral vertex covers, consider an arbitrary ℓ-tuple of integral vertex covers c 1 , . . . , c ℓ of W , and numbers γ 1 , . . . , γ ℓ ≥ 0 with γ i = 1. Consider an arbitrary i ∈ [ℓ]. We say that c i marks the set Proof of Claim 1. Suppose that this is not the case. Recall that k is odd. We can find an index h ∈ [k] that that A h and A h+1 are not marked (again, using the cyclic notation A k+1 = A k ). Therefore, the c i -preimages B h ⊆ A h and B h+1 ⊆ A h+1 of 0 have both measures more than α 4k . It follows from Lemma 11 and the way we set ε that W is positive on a set of positive measure on B h × B h+1 . This contradicts the fact that c i is a vertex cover.

Let us write
By convexity, we can replace A c i by A ( i γ i c i ) in (18). We now have Since neither the set A nor the bound on the right-hand side of (19) depend on the choice of the number ℓ, the vertex covers c i , and the constants γ i , we get that c is not in the weak * closure of convex combinations of integral vertex covers, as was needed.
3.6. An application: the Erdős-Gallai Theorem. In this section, we prove a graphon counterpart to the following classical result of Erdős and Gallai, [5].
Motivated by this, for e ∈ [0, 1] we define a graphons Ψ e and Φ e as a graphon as follows. We partition Ω = B 1 ⊔ B 2 so that ν(B 1 ) = 1 − √ 1 − e and ν(B 2 ) = √ 1 − e. We define Ψ e to be constant 0 on C 2 × C 2 and 1 elsewhere. We partition Ω = C 1 ⊔ C 2 so that ν(C 1 ) = e 2 and ν(C 2 ) = 1 − e 2 . We define Φ e to be constant 1 on C 1 × C 1 and 0 elsewhere. These definition uniquely determine Ψ e and Φ e , up to isomorphism. Thus, our graphon version of the Erdős-Gallai theorem reads as follows. This version of the Erdős-Gallai Theorem implies an asymptotic version of the finite statement. Furthermore, it provides a corresponding stability statement.
Furthermore, G contains a matching with at least ℓ + δn edges, unless G is εn 2 -close to the graph ExG(n, ℓ) as above in the edit distance.
The way of deriving Theorem 14 from Theorem 13 is standard, and we refer the reader to [6] where this was done in detail in the context of a tiling theorem of Komlós, [8], which is a statement of a similar flavor.
Let us emphasize that the original proof of Theorem 12 is simple and elementary (and the corresponding stability statement would not be difficult to prove with the same approach either). While our proof is not long, it makes use of the heavy machinery of graph limits, and in particular the results from [7] and from Section 3.2. However, we think that our proof offers an interesting alternative point of view on the problem.
For the proof of Theorem 13 we shall need the following fact. Proof. We transform this into an optimization problem in one variable by considering the function h(b) = g(2 (D − b), b). The function h is quadratic with limit plus infinity at −∞ and at +∞. Thus, the maximum of h on the interval [0, D] will be either at b = 0 or at b = D.
We have h(0) = 4D 2 and h(D) = 2D − D 2 . A quick calculation gives that the latter is bigger for D < 0.4 while the the latter is bigger for D > 0.4.

Convergence of polytons
4.1. Fractional vertex cover polytons of a convergent graphon sequence. Suppose that a sequence of graphons (W n ) n converges to a graphon W . We want to relate the polytons FCOV(W n ) to the polyton FCOV(W ). First, observe that the polytons FCOV(W n ) do not converge to FCOV(W ) in any reasonable sense in general. Indeed, for example, take W n to be a representation of a sample of the Erdős-Rényi random graph G(2n, 1/ log n). It is well-known that almost surely almost all these graphs contain a perfect matching. Thus, FCOV(W n ) contain only fractional vertex covers of size 1 2 and more. On the other hand, almost surely, the zero graphon W = 0 is the limit of (W n ) n , and so FCOV(W ) consists of all [0, 1]-valued measurable functions on Ω.
However, Theorem 16 below shows that FCOV(W ) asymptotically contains the polytons FCOV(W n ). This theorem is a special case of [7, Theorem 3.14].
Theorem 16. Suppose that (W n ) n is a sequence of graphons on Ω that converges to a graphon W : Ω 2 → [0, 1] in the cut-norm. Suppose that c n ∈ FCOV(W n ). Then any accumulation point of the sequence (c n ) n in the weak * topology lies in FCOV(W ).

4.2.
Matching polytons of a convergent graphon sequence. The main new result of this section concerns convergence properties of the matching polytons. This result is dual to Theorem 16: if W n converges to W then "MATCH(W n ) asymptotically contain MATCH(W )".
Theorem 17. Suppose that W : Ω 2 → [0, 1] is a graphon on a probability space Ω, and let m ∈ MATCH(W ) be fixed. Then for every ε > 0 there is δ > 0 such that whenever U : Ω 2 → [0, 1] is a graphon with U − W < δ then there is m U ∈ MATCH(U ) such that Since the cut-norm topology is stronger than the weak * topology, we get the following corollary.
Corollary 18. Suppose that (W n ) n is a sequence of graphons on a probability space Ω that converges to a graphon W : Ω 2 → [0, 1] in the cut-norm. Suppose that m ∈ MATCH(W ). Then there exists a sequence m n ∈ MATCH(W n ) such that (m n ) n converges to m in the cut-norm. In particular, the sequence (m n ) n converges to m in the weak * topology.
In the proof of Theorem 17, we will need the following technical lemma.
Moreover, it is obvious that such defined functionm is still a matching in the graphon W . We fixε > 0 such that Claim 2. There is r > 0 such that whenever Θ ⊆ Ω 2 is of positive measure such that Proof. By basic properties of measurable functions, there is s > 0 such that We will prove that r = 1 4Mε s works. Suppose for a contradiction that there is Θ ⊆ Ω 2 of positive measure with Then we have which is the desired contradiction with the definition of r.
Now we fix r > 0 from Claim 2, and we set By Lemma 19 there is a natural number k and a partition of Ω into pairwise disjoint subsets Ω 1 , . . . , Ω k , each of measure 1 k , such that where i, j = 1 . . . , n .
The first inequality from (27) easily implies that for all but at most ηk 2 pairs (i, j) we have Similarly, the second inequality from (27) implies that for all but at most ηk 2 pairs (i, j) we have We now set and we will show that this choice of δ works. So let U : Ω 2 → [0, 1] be a graphon such that denote the set of all those pairs (i, j) for which either (28) or (29) fails. We have that |A| ≤ 2ηk 2 . We define Claim 3. We have that t −m ≤ε.
Proof. We need to show that for every measurable sets S, T ⊆ Ω it holds So let us fix the sets S, T ⊆ Ω. Let Θ A denote the union of all the sets Ω i × Ω j for which (i, j) ∈ A. Similarly, let Θ B denote the union of all the sets Ω i × Ω j for which (i, j) / ∈ A and W ij < r, and let Θ C denote the union of all the sets Ω i × Ω j for which (i, j) / ∈ A and W ij ≥ r. The bulk of the work is in proving the following three subclaims.
Proof of Subclaim 1. Recall that by the definition of the set A, it holds ν ⊗2 (Θ A ) ≤ 2η, and so we have Proof of Subclaim 2. If ν ⊗2 (Θ B ) = 0 then trivially So suppose that Θ B is of positive measure. Note that then it clearly holds and so we have and consequently ν(B 1 ) < √ε . In the same way, we conclude that ν(B 2 ) < √ε . Now we are ready to define m U by setting ). Then we have So it remains to show that m U is a matching in the graphon U .
The fact that m U is a nonnegative function from L 1 (Ω 2 ) is obvious, and we also have supp (m U ) ⊆ supp (t) ⊆ supp (U ). So we only need to show that for almost every x ∈ Ω it holds y∈Ω m U (x, y) + y∈Ω m U (y, x) ≤ 1 .
This is trivially satisfied for every x ∈ B 1 ∪ B 2 as then the left-hand side of (32) equals 0. So let us fix x ∈ Ω \ (B 1 ∪ B 2 ). We may assume that y∈Ωm (x, y) + y∈Ωm (y, x) ≤ 1 , asm is a matching (in the graphon W which completes the proof of Theorem 17.

5.1.
Approximating MATCH(W ) and FCOV(W ) using W -random graphs. In Section 4, we showed that if W n → W then, in a certain sense, MATCH(W n ) asymptotically contain MATCH(W ), and FCOV(W n ) are asymptotically contained in FCOV(W ). We also showed that in general, these inclusions may be proper. However, we believe that we take W n as a representation of a typical W -random graph G(n, W ) then both these inclusions are asymptotically at equality.

5.2.
Bipartiteness from the matching polyton. Theorems 5 and 7 characterize bipartiteness of a graphon in terms of its fractional vertex cover polyton. For finite graphs there is another characterization in terms of the matching polytope: a graph is bipartite if and only if MATCH(G) is integral. Recall that there seems to be no counterpart to the concept of integrality of a graphon matching (c.f. Remark 3). So, we leave it as an important question to provide a characterization of bipartiteness in terms of MATCH(W ). It might be interesting to study this, and similar polytons. That said, let us emphasize that many basic results, like Edmonds' perfect matching polytope theorem, seem not to have a graphon counterpart as they concern integrality-related properties of the polytope.

5.4.
Generalizing the results to F -tilings. Results in Section 3 are specific to matchings -even in the finite setting. Even though we have not worked out details, we believe that our second main result, Theorem 17, extends to general F -tilings as introduced in [7] (and so does its proof).